Irodov Problem 1.148 – Momentum of System with Variable Mass

Problem Statement

Find the reactive force (thrust) on a rocket ejecting gas at speed u relative to itself, with mass ejection rate |dm/dt|.

Given Information

• Exhaust speed u
• Mass ejection rate |dm/dt|

Physical Concepts & Formulas

$$F_{thrust}=u\left|\frac{dm}{dt}\right|$$

Step-by-Step Solution

Step 1: Momentum change dP = m dv + v dm = 0 (external force free).

Step 2: Ejected mass dm at relative speed u gives impulse u|dm|.

Step 3: Thrust = u|dm/dt|.

Worked Calculation

F = u|dm/dt|

Answer

$$\boxed{F_{thrust}=u\left|\frac{dm}{dt}\right|}$$

Physical Interpretation

Thrust is simply the product of exhaust speed and mass flow rate. Higher exhaust velocity or faster burn rate both increase thrust independently.